3.2.2 Some examples

Name	Difference equations
Backward Euler	$w_j^{n+1}=w_j^n-\frac{\Delta t}{2\Delta x}\lambda(w_{j+1}^n-w_{j-1}^n)$
One-sided	$w_j^{n+1}=w_j^n-\frac{\Delta t}{\Delta x}\lambda(w_j^n-w_{j-1}^n)$
One-sided	$w_j^{n+1}=w_j^n-\frac{\Delta t}{\Delta x}\lambda(w_{j+1}^n-w_{j}^n)$
Lax-Friedrichs	$w_j^{n+1}=\frac{1}{2}(w_{j-1}^n+w_{j+1}^n)- \frac{\Delta t}{2\Delta x}\lambda(w_{j+1}^n-w_{j-1}^n)$
Leapfrog	$w_j^{n+1}=w_j^{n-1} - \frac{\Delta t}{2\Delta x}\lambda(w_{j+1}^n-w_{j-1}^n)$
Lax-Wendroff	$w_j^{n+1}=w_j^n-\frac{\Delta t}{2\Delta x}\lambda(w_{j+1}^n-w_{j-1}^n)$
	$+ \frac{(\Delta t)^2}{2(\Delta x)^2} \lambda^2 (w_{j+1}^n-2w_j^n+w_{j-1}^n)$
Beam-Warming	$w_j^{n+1}=w_j^n - \frac{\Delta t}{2\Delta x}\lambda(3w_j^n-4w_{j-1}^n+w_{j-2}^n)$
	$+ \frac{(\Delta t)^2}{2(\Delta x)^2} \lambda^2 (w_{j}^n-2w_{j-1}^n+w_{j-2}^n)$